question_answer

A rectangular glass slab ABCD of refractive index \[\frac{\pi }{6}\] is immersed in water of refractive index \[\frac{\pi }{2}\]A ray of light is incident at the surface AB of the slab as shown. The maximum value of the angle of incidence \[{{\cos }^{-1}}\frac{x}{2}+{{\cos }^{-1}}\frac{y}{3}=\theta ,\]such the ray comes out only from the another surface CD is given by

A) \[9{{x}^{2}}-12xy\cos 6+4{{y}^{2}}\]

B) \[-36{{\sin }^{2}}\theta \]

C) \[36{{\sin }^{2}}\theta \]                             

D) \[36{{\cos }^{2}}\theta \]

Correct Answer: A

Solution :

  Rays comes out only from CD, means rays after refraction from AB get totally internally reflected at AD. From the figure, \[3y-2+6=0\] \[y+3z+6=0\] \[3y-2z+6=0\]and \[a=2\hat{i}=2\hat{j}-2\hat{k}\] (for total internal reflection at AD) where, \[b=\hat{i}+\hat{j}\]or\[a.c=\left| c \right|,\left| c-a \right|=2\sqrt{2}\] \[a\times b\]\[\left| (a\times b)\times c \right|\] Now, applying Snell's law at face AB, we wet \[\frac{2}{3}\] \[\frac{3}{2}\] \[\frac{{{d}^{2}}y}{d{{x}^{2}}}={{\left\{ y+{{\left( \frac{dy}{dx} \right)}^{2}} \right\}}^{1/4}}\]\[1+\frac{2}{3}+\frac{6}{{{3}^{2}}}+\frac{10}{{{3}^{3}}}+\frac{14}{{{3}^{4}}}+...\] \[\frac{x}{2}-\frac{y}{3}=1\]